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Quantum error correction protects logical quantum information against environmental decoherence by encoding logical qubits into entangled states of physical qubits. One of the most important near-term challenges in building a scalable quantum computer is to reach the break-even point, where logical quantum circuits on error-corrected qubits achieve higher fidelity than equivalent circuits on uncorrected physical qubits. Using Quantinuum's H2 trapped-ion quantum processor, we encode the GHZ state in four logical qubits with fidelity $ 99.5 \pm 0.15 \% \le F \le 99.7 \pm 0.1\% $ (after postselecting on over 98% of outcomes). Using the same quantum processor, we can prepare an uncorrected GHZ state on four physical qubits with fidelity $97.8 \pm 0.2 \% \le F\le 98.7\pm 0.2\%$. The logical qubits are encoded in a $[\![ 25,4,3 ]\!]$ Tanner-transformed long-range-enhanced surface code. Logical entangling gates are implemented using simple swap operations. Our results are a first step towards realizing fault-tolerant quantum computation with logical qubits encoded in geometrically nonlocal quantum low-density parity check codes., Comment: 13 pages, 9 figures, 1 table. v2 changes: added classical simulation data and appendices

Code translation tools (transpilers) are developed for automatic source-to-source translation. Although learning-based transpilers have shown impressive enhancement against rule-based counterparts, owing to their task-specific pre-training on extensive monolingual corpora. Their current performance still remains unsatisfactory for practical deployment, and the associated training resources are also prohibitively expensive. LLMs pre-trained on huge amounts of human-written code/text have shown remarkable performance in many code intelligence tasks due to their powerful generality, even without task-specific training. Thus, LLMs can potentially circumvent the above limitations, but they have not been exhaustively explored yet. This paper investigates diverse LLMs and learning-based transpilers for automated code translation tasks, finding that: although certain LLMs have outperformed current transpilers, they still have some accuracy issues, where most of the failures are induced by a lack of comprehension of source programs, missing clear instructions on I/O types in translation, and ignoring discrepancies between source and target programs. Enlightened by the above findings, we further propose UniTrans, a Unified code Translation framework, applicable to various LLMs, for unleashing their power in this field. Specifically, UniTrans first crafts a series of test cases for target programs with the assistance of source programs. Next, it harnesses the above auto-generated test cases to augment the code translation and then evaluate their correctness via execution. Afterward, UniTrans further (iteratively) repairs incorrectly translated programs prompted by test case execution results. Extensive experiments are conducted on six settings of translation datasets between Python, Java, and C++. Three recent LLMs of diverse sizes are tested with UniTrans, and all achieve substantial improvements., Comment: 23 pages, 7 figures, accepted by FSE'24 (2024 ACM International Conference on the Foundations of Software Engineering)

Passive error correction protects logical information forever (in the thermodynamic limit) by updating the system based only on local information and few-body interactions. A paradigmatic example is the classical two-dimensional Ising model: a Metropolis-style Gibbs sampler retains the sign of the initial magnetization (a logical bit) for thermodynamically long times in the low-temperature phase. Known models of passive quantum error correction similarly exhibit thermodynamic phase transitions to a low-temperature phase wherein logical qubits are protected by thermally stable topological order. Here, in contrast, we show that certain families of constant-rate classical and quantum low-density parity check codes have no thermodynamic phase transitions at nonzero temperature, but nonetheless exhibit ergodicity-breaking dynamical transitions: below a critical nonzero temperature, the mixing time of local Gibbs sampling diverges in the thermodynamic limit. Slow Gibbs sampling of such codes enables fault-tolerant passive quantum error correction using finite-depth circuits. This strategy is well suited to measurement-free quantum error correction and may present a desirable experimental alternative to conventional quantum error correction based on syndrome measurements and active feedback., Comment: 32 pages, 9 figures, 1 table; v2 changes: added section on tree-codes, fixed typos

We constrain a broad class of teleportation protocols using insights from locality. In the "standard" teleportation protocols we consider, all outcome-dependent unitaries are Pauli operators conditioned on linear functions of the measurement outcomes. We find that all such protocols involve preparing a "resource state" exhibiting symmetry-protected topological (SPT) order with Abelian protecting symmetry $\mathcal{G}_{k}= (\mathbb{Z}_2 \times \mathbb{Z}_2)^k$. The $k$ logical states are teleported between the edges of the chain by measuring the corresponding $2k$ string order parameters in the bulk and applying outcome-dependent Paulis. Hence, this single class of nontrivial SPT states is both necessary and sufficient for the standard teleportation of $k$ qubits. We illustrate this result with several examples, including the cluster state, variants thereof, and a nonstabilizer hypergraph state., Comment: 34 pages, 8 figures; v2 changes: added summary of main assumptions and results

The surface code is a quantum error-correcting code for one logical qubit, protected by spatially localized parity checks in two dimensions. Due to fundamental constraints from spatial locality, storing more logical qubits requires either sacrificing the robustness of the surface code against errors or increasing the number of physical qubits. We bound the minimal number of spatially nonlocal parity checks necessary to add logical qubits to a surface code while maintaining, or improving, robustness to errors. We asymptotically saturate this bound using a family of hypergraph product codes, interpolating between the surface code and constant-rate low-density parity-check codes. Fault-tolerant protocols for logical gates in the quantum code can be inherited from its classical parent codes. We provide near-term practical implementations of this code for hardware based on trapped ions or neutral atoms in mobile optical tweezers. Long-range-enhanced surface codes outperform conventional surface codes using hundreds of physical qubits and represent a practical strategy to enhance the robustness of logical qubits to errors in near-term devices., Comment: 23 pages, 15 figures; v2 changes: fixed typos and added citations; v3 changes: updated numerics and new logical gadgets

Humans can categorize with only a few samples despite the numerous features. To mimic this ability, we propose a novel dimension-reduced category representation using a mixture of probabilistic principal component analyzers (mPPCA). Tests on the ${\tt CIFAR-10H}$ dataset demonstrate that mPPCA with only a single principal component for each category effectively predicts human categorization of natural images. We further impose a hierarchical prior on mPPCA to account for new category generalization. mPPCA captures human behavior in our experiments on images with simple size-color combinations. We also provide sufficient and necessary conditions when reducing dimensions in categorization is rational.

The speed of light $c$ sets a strict upper bound on the speed of information transfer in both classical and quantum systems. In nonrelativistic quantum systems, the Lieb-Robinson Theorem imposes an emergent speed limit $v \hspace{-0.2mm} \ll \hspace{-0.2mm} c$, establishing locality under unitary evolution and constraining the time needed to perform useful quantum tasks. We extend the Lieb-Robinson Theorem to quantum dynamics with measurements. In contrast to the expectation that measurements can arbitrarily violate spatial locality, we find at most an $(M \hspace{-0.5mm} +\hspace{-0.5mm} 1)$-fold enhancement to the speed $v$ of quantum information, provided the outcomes of measurements in $M$ local regions are known. This holds even when classical communication is instantaneous, and extends beyond projective measurements to weak measurements and other nonunitary channels. Our bound is asymptotically optimal, and saturated by existing measurement-based protocols. We tightly constrain the resource requirements for quantum computation, error correction, teleportation, and generating entangled resource states (Bell, GHZ, quantum-critical, Dicke, W, and spin-squeezed states) from short-range-entangled initial states. Our results impose limits on the use of measurements and active feedback to speed up quantum information processing, resolve fundamental questions about the nature of measurements in quantum dynamics, and constrain the scalability of a wide range of proposed quantum technologies., Comment: 24 pages main text + 61 pages SM; main text significantly expanded, includes new Lieb-Robinson bounds, additional discussion, more clarity on main results

We describe a simple quantum error correcting code built out of a time-dependent transverse field Ising model. The code is similar to a repetition code, but has two advantages: an $N$-qubit code can be implemented with a finite-depth spatially local unitary circuit, and it can subsequently protect against both $X$ and $Z$ errors if $N\ge 10$ is even. We propose an implementation of this code with 10 ultracold Rydberg atoms in optical tweezers, along with further generalizations of the code., Comment: 15 pages, 7 figures